We consider a directed polymer of length $L$ in a random medium of space dimension $d=1,2,3$. The statistics of low energy excitations as a function of their size $l$ is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities ${\rho }_L^{\mathit{\text{bulk}}}(E=0,l)$ and ${\rho }_L^{\mathit{\text{boundary}}}(E=0,l)$. We find that both densities follow the scaling behavior ${\rho }_L^{\mathit{\text{bulk}},\mathit{\text{boundary}}}(E=0,l)=L^{-1-{\theta }_d}{R}^{\mathit{\text{bulk}},\mathit{\text{boundary}}}(x=l∕L)$, where ${\theta }_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value ${\theta }_1=1∕3$ in one dimension). In the limit $x=l∕L\to 0$, both scaling functions $R^{\mathit{\text{bulk}}}\left(x\right)$ and $R^{\mathit{\text{boundary}}}\left(x\right)$ behave as $R^{\mathit{\text{bulk}},\mathit{\text{boundary}}}\left(x\right)\sim x^{-1-{\theta }_d}$, leading to the droplet power law ${\rho }_L^{\mathit{\text{bulk}},\mathit{\text{boundary}}}(E=0,l)\sim l^{-1-{\theta }_d}$ in the regime $1⪡l⪡L$. Beyond their common singularity near $x\to 0$, the two scaling functions $R^{\mathit{\text{bulk}},\mathit{\text{boundary}}}\left(x\right)$ are very different: whereas $R^{\mathit{\text{bulk}}}\left(x\right)$ decays monotonically for $0<x<1$, the function $R^{\mathit{\text{boundary}}}\left(x\right)$ first decays for $0<x<x_{\mathit{min}}$, then grows for $x_{\mathit{min}}<x<1$, and finally presents a power law singularity $R^{\mathit{\text{boundary}}}\left(x\right)\sim {(1-x)}^{-{\sigma }_d}$ near $x\to 1$. The density of excitations of length $l=L$ accordingly decays as ${\rho }_L^{\mathit{\text{boundary}}}(E=0,l=L)\sim L^{-{\lambda }_d}$ where ${\lambda }_d=1+{\theta }_d-{\sigma }_d$. We obtain ${\lambda }_1\simeq 0.67$, ${\lambda }_2\simeq 0.53$, and ${\lambda }_3\simeq 0.39$, suggesting the possible relation ${\lambda }_d=2{\theta }_d$.

We investigate the quantum dynamics of a periodically kicked Bose-Einstein condensate (BEC) confined in a one-dimensional (1D) box both numerically and theoretically, emphasizing on the phenomena of quantum resonance and antiresonance. The quantum resonant behavior of BEC is different from the single particle case but the antiresonance condition ($T=2\pi$ and $\alpha =0$) is not affected by the atomic interaction. For the antiresonance case, the nonlinearity (atom interaction) causes the transition between oscillation and quantum beating. For the quantum resonance case, because of the coherence of BEC, the energy increase is oscillating and the rate is dramatically affected by the many-body interaction. We also discuss the relation between the quantum resonant behavior and the Kolmogorov-Arnold-Moser (KAM) or non-KAM property of the corresponding classical system.

This paper provides a unified treatment of the fundamental methods used to obtain the density of states $\omega$ via molecular simulations with isothermal ensembles (IEs) and adiabatic ensembles (AEs). Our analysis and results show that $\omega$ provides a natural bridge to go back and forth between IE and AE simulation data. They also underline the difference between the density of states of potential energy macrostates $\omega$ and that of total energy macrostates $\Omega$, even though both provide access to the thermodynamic properties of the system. Visited-states approaches and transition matrix methods are described and applied to the Lennard-Jones fluid to target $\omega$ and $\Omega$ as functions of energy and volume macrostates. It is shown that one can obtain $\omega$ via a generalized acceptance-ratio formula that is applicable regardless of the conditions at which the ensemble is simulated. In this way, one can obtain $\omega$ while performing conventional IE or AE simulations, and do it at no extra cost and with a higher accuracy than is achievable with histogram methods.

We obtained the global persistence exponent ${\theta }_g$ for a continuous spin model on the simple cubic lattice with double-exchange interaction by using two different methods. First, we estimated the exponent ${\theta }_g$ by following the time evolution of probability $P\left(t\right)$ that the order parameter of the model does not change its sign up to time $t\left[P\right(t)\sim t^{-{\theta }_g}]$. Afterwards, that exponent was estimated through the scaling collapse of the universal function $L^{{\theta }_{g}z}P\left(t\right)$ for different lattice sizes. Our results for both approaches are in very good agreement with each other.

We discuss the exact solution for the properties of the recently introduced “necklace” model for reptation. The solution gives the drift velocity, diffusion constant, and renewal time for asymptotically long chains. Its properties are also related to a special case of the Rubinstein-Duke model in one dimension.

In higher organisms, DNA replicates simultaneously from many origins. Recent in vitro experiments have yielded large amounts of data on the state of replication of DNA fragments. From measurements of the time dependence of the average size of replicated and nonreplicated domains, one can estimate the rate of initiation of DNA replication origins, as well as the average rate at which DNA bases are copied. One problem in making such estimates is that, in the experiments, the DNA is broken up into small fragments, whose finite size can bias downward the measured averages. Here, we present a systematic way of accounting for this bias by deriving theoretical relationships between the original domain-length distributions and fragment-domain length distributions. We also derive unbiased average-domain-length estimators that yield accurate results, even in cases where the replicated (or nonreplicated) domains are larger than the average DNA fragment. Then we apply these estimators to previously obtained experimental data to extract improved estimates of replication kinetics parameters.

Earthquake phenomenology exhibits a number of power law distributions including the Gutenberg-Richter frequency-size statistics and the Omori law for aftershock decay rates. In search for a basic model that renders correct predictions on long spatiotemporal scales, we discuss results associated with a heterogeneous fault with long-range stress-transfer interactions. To better understand earthquake dynamics we focus on faults with Gutenberg-Richter-like earthquake statistics and develop two universal scaling functions as a stronger test of the theory against observations than mere scaling exponents that have large error bars. Universal shape profiles contain crucial information on the underlying dynamics in a variety of systems. As in magnetic systems, we find that our analysis for earthquakes provides a good overall agreement between theory and observations, but with a potential discrepancy in one particular universal scaling function for moment rates. We primarily use mean field theory for the theoretical analysis, since it has been shown to be in the same universality class as the full three-dimensional version of the model (up to logarithmic corrections). The results point to the existence of deep connections between the physics of avalanches in different systems.

We propose a new autonomous dynamical system of dimension $N=4$ that demonstrates the regime of stable two-frequency motions and period-doubling bifurcations of a two-dimensional torus. It is shown that the period-doubling bifurcation of the two-dimensional torus is not followed by the resonance phenomenon, and the two-dimensional ergodic torus undergoes a period-doubling bifurcation. The interaction of two generators is also analyzed. The phenomenon of external and mutual synchronization of two-frequency oscillations is observed, for which winding number locking on a two-dimensional torus takes place.

Drift-wave turbulence control in a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave is considered. We apply time-delay and space-shift feedback signals, to suppress turbulence. By using global and local pinning strategies, we show numerically that the turbulent state can be controlled to periodic states effectively if appropriate time-delay length and space-shift distance are chosen. The physical mechanism of the control scheme is understood based on the energy-minimum principle.

The replica method has played a key role in analyzing systems with disorder, e.g., the Sherrington-Kirkpatrick (SK) model, and associative neural networks. Here we study the influence of disorder in the lateral inhibition type interactions on the cooperative and uncooperative behavior of recurrent neural networks by using the replica method. Although the interaction between neurons has a dependency on distance, our model can be solved analytically. Bifurcation analysis identifies the boundaries between paramagnetic, ferromagnetic, spin-glass, and localized phases. In the localized phase, the network shows a bump like activity, which is often used as a model of spatial working memory or columnar activity in the visual cortex. Simulation results show that disordered interactions can stabilize the drift the of bump position, which is commonly observed in conventional lateral inhibition type neural networks.

We simulate the tapping of a bed of hard disks in a rectangular box by using a pseudodynamic algorithm. In these simulations, arches are unambiguously defined and we can analyze their properties as a function of the tapping amplitude. We find that an order-disorder transition occurs within a narrow range of tapping amplitudes as has been seen by others. Arches are always present in the system although they exhibit regular shapes in the ordered regime. Interestingly, an increase in the number of arches does not always correspond to a reduction in the packing fraction. This is in contrast with what is found in three-dimensional systems.

It has been suggested that thermally induced separations (“bubbles”) of the DNA double-strand may play a role in the initiation of gene transcription, and an accurate understanding of the sequence dependence of thermal strand separation is therefore desirable. Based on the Peyrard-Bishop-Dauxois model, we show here that the bubble forming ability of DNA can be quantified in terms of a healing length $L\left(n\right)$, defined as the length (number of base-pairs) over which a base-pair defect affects bubbles involving $n$ consecutive base-pairs. The probability for a bubble of size $n$ is demonstrated to be proportional to the number of adenine-thymine base-pairs found within this length. The method for calculating bubble probabilities in a given sequence derived from this notion requires several order of magnitude less numerical effort than direct evaluation.

Microsecond relaxation dynamics in a cavitating surface layer of bulk water superheated by a TEA $\mathrm{C}{\mathrm{O}}_2$ laser was studied using contact broadband photoacoustic spectroscopy. Damped nanosecond and microsecond oscillatory pressure-tension cycles recorded by an acoustic transducer are related to oscillations of steam bubbles of different sizes exhibiting strong dissipative losses and collective (coalescence and percolation) phenomena. These measurements also give important insight into basic parameters, characteristic spatial and temporal scales, and the mechanism of laser ablation of absorbing liquids in the thermal confinement regime.

Modulation theory is developed for a periodic peakon solution of the Camassa-Holm equation. An explicit simple wave solution of these modulation equations is then derived; this simple wave describing the evolution into an undular bore of an initial step. The characteristic on which the expansion fan occurs (propagating at a nonlinear group velocity) has a turning point, illustrating the fact that there is a minimum nonlinear group velocity at which the waves can propagate. A linear analytical solution, based on an integral of the Airy function, is then derived to describe the evanescent portion of the undular bore behind the turning point. Good agreement is found between the modulation theory plus Airy integral solution and numerical solutions.

We investigate the nonlinear oscillations of heat-conductive, viscous, liquid drops in vacuum with zero gravity, using smoothed particle hydrodynamics (SPH). The liquid drops are modeled as a van der Waals fluid in two dimensions so that the models apply to flat, disklike drops. Attention is focused on small- to large-amplitude oscillations of drops that are released from a static elliptic shape. We find that for small-amplitude motions the combined dissipative effects of finite viscosity and heat conduction induce rapid decay of the oscillations after a few periods, while for large-amplitude motions wave damping is governed by the action of both viscous dissipation and surface tension forces. The transition from periodic to aperiodic decay at $\mathrm{Re}\sim 1$ as well as the quadratic decrease of the frequency with the initial aspect ratio at large Re are reproduced in good agreement with previous theoretical predictions and experimental results.

The dielectric measurements have been carried out on the two zeroth generation dendrimers with four branched arms (called tetrapodes) based on the siloxane cores. The results are analyzed in the framework of the molecular theory of dielectric permittivity by Maier and Meier for nematogens. At least four molecular processes are resolved in the dielectric relaxation spectra in the nematic phase for each of the two tetrapodes. Three of them are assigned to the reorientation of the monomeric unit whereas the fourth is assigned to the rotation of the molecular segments in the individual arms of the monomeric unit around the long molecular axis. The dielectric relaxation strength of the low frequency process has been used to calculate the orientational order parameter. The dynamics of the resolved processes has been quantitatively analyzed using the results of the microscopic model of the rotational diffusion, given by Coffey and Kalmykov [W. T. Coffey and Yu. P. Kalmykov, Adv. Chem. Phys. 113, 487 (2000)] using the calculated order parameter. All molecular processes: the rotation around the short molecular axis (end-over-end rotation), precession around the director and the rotation around the long molecular axis (also called the spinning motion) are shown to have successfully been reproduced by the model. The anisotropy of the rotational diffusion coefficients gradually increases with a reduction in temperature, to a factor of 3 in the nematic phase relative to its isotropic phase.

We investigate condensation phase transitions of the symmetric conserved-mass aggregation (SCA) model on random networks (RNs) and scale-free networks (SFNs) with degree distribution $P\left(k\right)\sim k^{-\gamma }$. In the SCA model, masses diffuse with unit rate, and unit mass chips off from mass with rate $\omega$. The dynamics conserves total mass density $\rho$. In the steady state, on RNs and SFNs with $\gamma >3$ for $\omega \ne \infty$, we numerically show that the SCA model undergoes the same type of condensation transitions as those on regular lattices. However, the critical line ${\rho }_c\left(\omega \right)$ depends on network structures. On SFNs with $\gamma ⩽3$, the fluid phase of exponential mass distribution completely disappears and no phase transitions occurs. Instead, the condensation with exponentially decaying background mass distribution always takes place for any nonzero density. For the existence of the condensed phase for $\gamma ⩽3$ at the zero density limit, we investigate one lamb-lion problem on RNs and SFNs. We numerically show that a lamb survives indefinitely with finite survival probability on RNs and SFNs with $\gamma >3$, and dies out exponentially on SFNs with $\gamma ⩽3$. The finite lifetime of a lamb on SFNs with $\gamma ⩽3$ ensures the existence of the condensation at the zero density limit on SFNs with $\gamma ⩽3$, at which direct numerical simulations are practically impossible. At $\omega =\infty$, we numerically confirm that complete condensation takes place for any $\rho >0$ on RNs. Together with the recent study on SFNs, the complete condensation always occurs on both RNs and SFNs in zero range process with constant hopping rate.

We study the probability density function $P(h_m,L)$ of the maximum relative height $h_m$ in a wide class of one-dimensional solid-on-solid models of finite size $L$. For all these lattice models, in the large-$L$ limit, a central limit argument shows that, for periodic boundary conditions, $P(h_m,L)$ takes a universal scaling form $P(h_m,L)\sim {\left(\sqrt{12}{w}_L\right)}^{-1}f\mathbf{(}{h}_m∕\left(\sqrt{12}{w}_L\right)\mathbf{)}$, with $w_L$ the width of the fluctuating interface and $f\left(x\right)$ the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using the transfer matrix technique, valid for any $L>0$. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a nonuniversal amplitude, and simply given by the derivative of the Airy distribution function $f^{\prime }\left(x\right)$.

We show that the nontwist phenomena previously observed in Hamiltonian systems exist also in time-reversible non-Hamiltonian systems. In particular, we study the two standard collision-reconnection scenarios and we compute the parameter space breakup diagram of the shearless torus. Besides the Hamiltonian routes, the breakup may occur due to the onset of attractors. We study these phenomena in coupled phase oscillators and in non-area-preserving maps.

Based on the model proposed by Hilgenfeldt et al. [Nature (London) 398, 401 (1999)], we present here a comprehensive theory of thermal radiation in single-bubble sonoluminescence (SBSL). We first invoke the generalized Kirchhoff’s law to obtain the thermal emissivity from the absorption cross section of a multilayered sphere (MLS). A sonoluminescing bubble, whose internal structure is determined from hydrodynamic simulations, is then modeled as a MLS and in turn the thermal radiation is evaluated. Numerical results obtained from simulations for argon bubbles show that our theory successfully captures the major features observed in SBSL experiments.